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Zbl 0828.22019
Aslaksen, Helmer; Tan, Eng-Chye; Zhu, Chen-Bo
On certain rings of highest weight vectors. (English)
[J] J. Algebra 174, No.1, 159-186 (1995). ISSN 0021-8693

In this paper the authors consider the representation of the group $G = O_m \times GL_n$ on the polynomial algebra $P(V)$ of $m \times n$- matrices induced by the natural action by left, resp., right multiplication on $V = \bbfC^{m,n}$. Let $B = UT$ be a Borel subgroup of $G$, where $T$ is a maximal torus and $U$ a maximal unipotent subgroup. Then $T$ acts semisimply on the space $P(V)^U$ of $U$- invariants in $P(V)$. The eigenvectors for this representation of $T$ are called highest weight vectors and $R_{m,n} := P(V)^U$ the ring of highest weight vectors. The main objective of the authors is to study the structure of this ring. One major tool to do this is the decomposition $P(V) = P(V)^{O_m} \cdot {\cal H}$, where ${\cal H} = \{f : (\forall 1 \leq j \leq k \leq n) \Delta_{j,k} \cdot f = 0\}$ is the space of harmonics, and $\Delta_{jk} = \sum^m_{s = 1} {\partial^2 \over \partial x_{sj} \partial x_{sk}}$. For $m \geq 3$ this leads directly to the description of $R_{m,1}$ as $\bbfC [z_1, r^2_{11}]$ with $z_1 = x_{11} - ix_{21}$ and $r^2_{jk} = \sum^m_{s = 1} x_{sj} x_{sk}$. Another important source of information is the work of {\it M. Kashiwara} and {\it M. Vergne} [Invent. Math. 44, 1-47 (1978; Zbl 0375.22009)]. In the article under review the authors determine the ring $R_{m,2}$ and use it to compute for the holomorphic representations of $\text {Sp}_4$ the Poincaré series, the Gelfand- Kirillov dimension and the Berenstein degree. They also describe a set of generators for $R_{m,3}$.
[K.-H.Neeb (Erlangen)]

MSC 2000:: *22E46 Semi-simple Lie groups and their representations
20G05 Representation theory of linear algebraic groups
22E30 Analysis on real and complex Lie groups
22E45 Analytic repres.of Lie and linear algebraic groups over real fields

Keywords: representation; Borel subgroup; eigenvectors; highest weight vectors

Citations: Zbl 0375.22009

Cited in: Zbl 0919.22005

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